Invariant measures on homogeneous spaces, with applications to function spaces and lattice counting

TL;DR

This paper studies invariant measures on homogeneous spaces of real reductive groups, characterizes smooth vectors in associated Banach representations, and applies these results to lattice counting problems.

Contribution

It establishes measure estimates on G/H and characterizes when smooth vectors vanish at infinity, linking reductive type to functional properties.

Findings

Smooth vectors in L^p(G/H) vanish at infinity iff G/H is of reductive type.

Provides measure estimates for invariant measures on G/H.

Applies results to lattice counting on homogeneous spaces.

Abstract

Let G be a real reductive group and G/H a unimodular homogeneous G space with a closed connected subgroup H. We establish estimates for the invariant measure on G/H. Using these, we prove that all smooth vectors in the Banach representation L^p(G/H) of G are functions that vanish at infinity if and only if G/H is of reductive type. An application to lattice counting on G/H is presented.